Involve: A Journal of Mathematics

  • Involve
  • Volume 11, Number 3 (2018), 489-500.

A tale of two circles: geometry of a class of quartic polynomials

Christopher Frayer and Landon Gauthier

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Let P be the family of complex-valued polynomials of the form p(z)=(z1)(zr1)(zr2)2 with |r1|=|r2|=1. The Gauss–Lucas theorem guarantees that the critical points of pP will lie within the unit disk. This paper further explores the location and structure of these critical points. For example, the unit disk contains two “desert” regions, the open disk {z:|z34|<14} and the interior of 2x43x3+x+4x2y23xy2+2y4=0, in which critical points of p cannot occur. Furthermore, each c inside the unit disk and outside of the two desert regions is the critical point of at most two polynomials in P.

Article information

Involve, Volume 11, Number 3 (2018), 489-500.

Received: 21 February 2017
Revised: 5 June 2017
Accepted: 13 June 2017
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}

geometry of polynomials critical points Gauss–Lucas theorem


Frayer, Christopher; Gauthier, Landon. A tale of two circles: geometry of a class of quartic polynomials. Involve 11 (2018), no. 3, 489--500. doi:10.2140/involve.2018.11.489.

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